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Noether Theorems in a General Setting. Reducible Graded Lagrangians

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Noether Theorems in a General Setting. Reducible Graded Lagrangians International Conference ”Geometry of Jets and Fields” 10-16 May 2015, Bedlewo, Poland on the 60th birthday of Janusz Grabowski G. Sardanashvily Noether theorems in a general setting. Reducible graded Lagrangians Basic reference: G.Sardanashvily, Noether theorems in a general setting, arXiv: 1411.2910. Main publications on the subject: D.Bashkirov, G.Giachetta, L.Mangiarotti, G.Sardanashvily, Noether’s second theorem in a general setting. Reducible gauge theory, J. Phys. A 38 (2005) 5329. D.Bashkirov, G.Giachetta, L.Mangiarotti, G.Sardanashvily, The antifield Koszul – Tate com- plex of reducible Noether identities, J. Math. Phys. 46 (2005) 103513. G.Giachetta, L.Mangiarotti, G.Sardanashvily, Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology, Commun. Math. Phys. 259 (2005) 103. G.Sardanashvily, Noether identities of a differential operator. The Koszul – Tate complex, Int. J. Geom. Methods Mod. Phys. 2 (2005) 873. D.Bashkirov, G.Giachetta, L.Mangiarotti, G.Sardanashvily, The KT-BRST complex of de- generate Lagrangian systems, Lett. Math. Phys. 83 (2008) 237. G.Giachetta, L.Mangiarotti, G.Sardanashvily, On the notion of gauge symmetries of generic Lagrangian field theory, J. Math. Phys 50 (2009) 012903. G.Sardanashvily, Graded Lagrangian formalism, Int. J. Geom. Methods. Mod. Phys. 10 (2013) 1350016. 1 Main Theses Here we are not concerned with the rich history of Noether theorems, and refer for this subject to a brilliant book: Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and the Conservation Laws in the Twentieth Century (Springer, 2011). • Second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. • Such Lagrangian theory is characterized by a hierarchy of non-trivial Noether and higher-stage Noether identities which are described as elements of homology groups of some chain complex. • If a certain homology regularity condition holds, one can associate to a reducible degenerate Lagrangian the exact Koszul – Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete non-trivial Noether and higher-stage Noether identities. • Second Noether theorems associate to the above-mentioned Koszul – Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. • If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above mentioned cochain sequence into the BRST complex and thus provides a BRST extension of an original Lagrangian system. 2 Problems • A key problem is that any Euler – Lagrange operator satisfies Noether identities, which therefore must be separated into the trivial and non-trivial ones. • These Noether identities can obey first-stage Noether identities, which in turn are subject to the second-stage ones, and so on. Thus, there is a hierarchy of Noether identities and higher-stage Noether identities which also must be separated into the trivial and non-trivial ones • This hierarchy of Noether and higher-stage Noether identities is de- scribed in the framework of a Grassmann-graded homology complex. There- fore Lagrangian theory of Grassmann-graded even and odd variables should be considered from the beginning. 3 Graded Lagrangian formalism Lagrangian theory of even variables on a smooth manifold X, dim X = n > 1, conventionally is formulated in terms of fibre bundles and jet mani- folds. However, different geometric models of odd variables either on graded manifolds or supermanifolds are discussed. Both graded manifolds and super- manifolds are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces. • We follow the Serre – Swan theorem extended to graded manifolds. It states that, if a graded commutative C∞(X)-ring is generated by a pro- jective C∞(X)-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. Since higher-stage Noether identities of a Lagrangian system on a manifold X form graded C∞(X)- modules, we describe odd variables in terms of graded manifolds. • A graded manifold is characterized by a body manifold Z and a structure sheaf A of Grassmann algebras on Z. Its sections form a graded commuta- tive structure C∞(Z)-ring A of graded functions on (Z, A). The differential calculus on a graded manifold is the Chevalley – Eilenberg differential calculus over its structure ring. By virtue of Batchelor’s theorem, there is a vector bundle E → Z such that a structure ring of (Z, A) is isomorphic ∗ ∗ to a ring AE of sections of an exterior bundle ∧E of the dual E of E. This isomorphism is not canonical, but usually is fixed from the beginning. Therefore, we restrict our consideration to graded manifolds (Z, AE), called the simple graded manifolds, modelled over vector bundles E → Z. 4 • A common configuration space of even and odd variables is defined as a graded bundle (Y, AF ) which is a simple graded manifold modelled over a vector bundle F → Y whose body is Y . Its graded r-order jet manifold r (J Y, Ar = AJ rF ) is introduced as a simple graded manifold modelled over a vector bundle J rF → J rY . This definition of graded jet manifolds is compatible with the conventional one of jets of fibre bundles. • Lagrangian theory on a fibre bundle Y adequately is phrased in terms of the variational bicomplex of exterior forms on the infinite order jet manifold J ∞Y of Y . This technique is extended to Lagrangian theory on graded manifolds in terms of the Grassmann-graded variational bicomplex ∗ S∞[F ; Y ] of graded exterior forms on a graded infinite order jet manifold ∞ r (J Y, AJ ∞F ) which is the inverse limit of graded manifolds (J Y, Ar). • Lagrangians L and the Euler – Lagrange operator δL are defined as the elements and the coboundary operator of this graded bicomplex. Its cohomology provides the global variational formula n−1 dL = δL − dH ΞL, ΞL ∈ S∞ [F ; Y ],, (1) where ΞL is a global Lepage equivalent of a graded Lagrangian L. ∗ • Given a graded Lagrangian system (S∞[F ; Y ],L), by its infinitesimal transformations are meant contact graded derivations ϑ of a graded com- 0 mutative ring S∞[F ; Y ]. Every graded derivation ϑ yields a Lie derivative ∗ Lϑ of a graded algebra S∞[F ; Y ]. It is the infinite order jet prolongation ϑ = J ∞υ of its restriction υ to a graded commutative ring S0[F ; Y ]. 5 Noether identities We follow the general analysis of Noether identities (NI) and higher-stage NI of differential operators on fibre bundles when trivial and non-trivial NI are represented by boundaries and cycles of a chain complex. ∗ • One can associate to any graded Lagrangian system (S∞[F ; Y ],L) the chain complex (2) whose one-boundaries vanish on the shell δL = 0. Let us n consider the density-dual VF = V ∗F ⊗ ∧ T ∗X → F , of the vertical tangent F ∗ bundle VF → F , and let us enlarge an original algebra S∞[F ; Y ] with the A ∗ A generating basis (s ) to P∞[VF ; Y ] with the generating basis (s , sA), [sA] = [A] + 1. Following the terminology of BRST theory, we call its elements sA ∗ the antifields of antifield number Ant[sA] = 1. An algebra P∞[VF ; Y ] is ← A endowed with the nilpotent right graded derivation δ =∂ EA, where EA = δAL are the variational derivatives. Then we have a chain complex δ 0,n δ 0,n 0←Im δ ←−P∞ [VF ; Y ]1 ←−P∞ [VF ; Y ]2 (2) of graded densities of antifield number ≤ 2. Its one-boundaries δΦ, Φ ∈ 0,n P∞ [VF ; Y ]2, by very definition, vanish on-shell. Any one-cycle Φ of the complex (2) is a differential operator on a bundle VF such that its kernel contains the graded Euler – Lagrange operator δL, i.e., X A,Λ δΦ = 0, Φ dΛEAω = 0, (3) 0≤|Λ| where dΛ are total derivatives relative to a multi-index Λ = λ1 ··· λk. Refer- eing to a notion of NI of a differential operator, we say that the one-cycles Φ define the Noether identities (3) of an Euler – Lagrange operator δL. 6 • One-chains Φ are necessarily NI if they are boundaries. Therefore, these NI are called trivial. Accordingly, non-trivial NI modulo trivial ones are associated to elements of the first homology H1(δ) of the complex (2). A Lagrangian L is called degenerate if there are non-trivial NI. • Non-trivial NI can obey first-stage NI. To describe them, let us assume that a module H1(δ) is finitely generated. Namely, there exists a graded ∞ projective C (X)-module C(0) ⊂ H1(δ) of finite rank possessing a local basis {∆rω} such that any element Φ ∈ H1(δ) factorizes as X r,Ξ r,Ξ 0 Φ = Φ dΞ∆rω, Φ ∈ S∞[F ; Y ], (4) 0≤|Ξ| through elements ∆rω of C(0). Thus, all non-trivial NI (3) result from the NI X A,Λ δ∆r = ∆r dΛEA = 0, (5) 0≤|Λ| called the complete NI. Then the complex (2) can be extended to the chain complex (6) with a boundary operator whose nilpotency is equivalent to the complete NI (5). By virtue of Serre – Swan theorem, a graded module C(0) is isomorphic to that of sections of the density-dual E0 of some ∗ graded vector bundle E0 → X. Let us enlarge P∞[VF ; Y ] to an algebra ∗ ∗ A P∞{0} = P∞[VF × E0; Y ] with a generating basis (s , sA, cr) where cr are X antifields such that [cr] = [∆r] + 1 and Ant[cr] = 2.
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